Solve each equation. Give solutions in exact form. ln x + ln x² = 3

Verified step by step guidance

1. Recall the logarithm property that allows you to combine sums of logarithms with the same base: \(\ln a + \ln b = \ln (a \cdot b)\). Apply this to the left side of the equation \(\ln x + \ln x^{2}\) to combine the terms into a single logarithm.
2. Using the property, rewrite the equation as \(\ln (x \cdot x^{2}) = 3\). Simplify the product inside the logarithm to get \(\ln (x^{3}) = 3\).
3. Recall that \(\ln y = c\) is equivalent to the exponential form \(y = e^{c}\). Use this to rewrite \(\ln (x^{3}) = 3\) as \(x^{3} = e^{3}\).
4. To solve for \(x\), take the cube root of both sides of the equation: \(x = \sqrt[3]{e^{3}}\). This can also be written as \(x = e^{3/3}\) by using the property of exponents \(\sqrt[n]{a^{m}} = a^{m/n}\).
5. Simplify the exponent to get \(x = e^{1}\). Remember to check the domain of the original logarithmic expressions to ensure the solution is valid (i.e., \(x > 0\)).

Key Concepts

Properties of Logarithms

Logarithmic properties, such as the product rule ln(a) + ln(b) = ln(ab), allow combining or expanding logarithmic expressions. Understanding these rules is essential to simplify equations involving multiple logarithms into a single logarithm for easier solving.

Solving Exponential Equations

After rewriting logarithmic equations in exponential form, solving for the variable involves isolating the base raised to a power. This process converts the logarithmic equation into a polynomial or algebraic equation that can be solved using standard algebraic methods.

Domain Restrictions of Logarithmic Functions

Logarithms are only defined for positive arguments. When solving equations involving ln(x), it is crucial to consider domain restrictions to exclude any solutions that make the argument zero or negative, ensuring all solutions are valid.